Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(9,10)(11,15)(12,16)(13,17)(14,20)(18,22)(19,23)(21,24);(2,4)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22);(10,16)(12,20)(14,17)(15,22)(18,24)(21,23);(10,17)(14,16)(15,23)(21,22)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72249507, #72249565, #72249594, … |
| maximal normal subgroup of | #72249507, #72249565, #72249594, … |
| maximal quotient of | #72264411, #72264418, #72264502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286344);